If planets were connected with "space elevators" lol. Would it still take just as long to get to another planet?

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Not necessarily, although it could also take *longer* because you could potentially accelerate at a much faster rate using other methods.
A space elevator is not a realistic method of traveling between planets, a space elevator merely gets you into orbit. I say 'merely' because its the shorter part of the trip in terms of time but can easily be the most costly in terms of energy, it takes a lot of energy to get into orbit using a rocket.

Xylene, you dont think that the cable (and anyone ON the elivator at the time) going straight through the sun (and or any other objects which happened to get in the way like other planets) would be a significant problem?

Of course it would make much more sense to have space elevators between parent planet and moon (e.g. Pluto and Charon) rather than between different planets, because of the constant change in relative position around the Sun. Alternatively SE's could be between twin planets that were tidally locked in a mutual orbit, i.e. going around the sun together as a constant pair. That's the only way you could prevent your scenario.

Of course it would make much more sense to have space elevators between parent planet and moon (e.g. Pluto and Charon) rather than between different planets, because of the constant change in relative position around the Sun. Alternatively SE's could be between twin planets that were tidally locked in a mutual orbit, i.e. going around the sun together as a constant pair. That's the only way you could prevent your scenario.

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ok great, so you now have an elivator between the earth and the moon at a cost of however many billion $. Now an asteroid happens to fly between the earth and the moon. Ops, no more elivator, oh dear all that money down the drain. This is a much less likly issue if the elivator was going from the ground to a space station in orbit because its a shorter cable so less likly to get hit.

Second problem, the moon moves in the sky. "Oh no, whats that huge cable wrapping itself around the earth?" and then either "snap" or we have the moon crash into earth. Not a good thing

The only concivable way to do this is if both objects were in geostationary orbit relitive to the other one so that nither ever changed in the sky (face OR position) and even if this was the case you are still streching a dirty great big wire (which cant be moved out of the way of an asterioid like a ship can) which can potentually be hit by any asteroids which pass between them

ok great, so you now have an elivator between the earth and the moon at a cost of however many billion $. Now an asteroid happens to fly between the earth and the moon. Ops, no more elivator, oh dear all that money down the drain. This is a much less likly issue if the elivator was going from the ground to a space station in orbit because its a shorter cable so less likly to get hit.

Second problem, the moon moves in the sky. "Oh no, whats that huge cable wrapping itself around the earth?" and then either "snap" or we have the moon crash into earth. Not a good thing

The only concivable way to do this is if both objects were in geostationary orbit relitive to the other one so that nither ever changed in the sky (face OR position) and even if this was the case you are still streching a dirty great big wire (which cant be moved out of the way of an asterioid like a ship can) which can potentually be hit by any asteroids which pass between them

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OK, fair points; here are some thoughts;
1) Build a whole series of modules that are linked together, connected with universal joints like a set of carriages on a train. Such a line would be (hopefully) more flexible, able to be shifted out of the way of a potential asteroid strike.

2) If an asteroid strike occurs, two or three of the modules might be broken up by the impact, but could be replaced and the line reconnected.

3) Clean up the space around Earth, getting rid of all the space junk that's currently circulating (which is far more likely to cause chaos than an asteroid) and also clean up the asteroid population around the moon (ongoing job, should soak up a bit of unemployment on Earth).

5) No need to attach the tether to the planetary surface; the end could placed in LEO, where it could be accessed by shuttle craft at either end. Otherwise, if you have the tether literally tethered to either planetary surface, you're always going to be allowing for the possiblity of some nutjob blowing it up or otherwise creating mayhem.

Xylene, there is no point. The point of a space elevator is that when someone wants to go up you send something down which provides the nessary energy to send the people up (like a see-saw). If your doing it between orbiting bodies it wont work, you need a gravitational effect or the whole thing is pointless. Once in orbit space ships are quite efficent because you only need to give them a shove in one direction and they will keep going till you give them a shove back so if your going in a straight line you only need to fire the rockets twice. Its the amount of fuel expended REACHING orbit that is the issue and thats where the elevator comes in.

Example:

Miners mine stuff on moon, send it up the space elevator in exhange for surplies from earth and possably using a little power (not so big an issue as the moons gravity is low)

Ship flys from space station in geostationary orbit above moon to space station in geostationary orbit around earth and unloads ore and picks up more surplies\ people and fuel to fly back to moon

ore goes down the elevator to earth and surplies\people\fuel goes up (or you could possably double up and use an electric power source on the SS to fuel the rocket and collect that power at the SS)

If you use a rocket to go up from the moon and from the earth and then just use a space elevator for the EASY part of the journey then you are wasting money